“Dysfunction in this market, and the prospect of self-reinforcing ‘fire sale’ dynamics pose a material risk to U.K. financial stability,” the BOE said.

The yield on a 30-year U.K. inflation-linked bond has soared above 1.5% this week, up from 0.851% on Oct. 7, according to

Tradeweb.

Just weeks ago, the yield on the gilt, as U.K. government bonds are known, was negative. Because yields rise as prices fall, the effect has been punishing losses for bond investors.

On Tuesday, after the BOE expanded the purchases, the yield on inflation-linked gilts held mostly steady but at the new, elevated levels. The central bank said it bought roughly £2 billion, equivalent to about $2.21 billion, in inflation-linked gilts, out of a £5 billion daily capacity.

The bank’s bond purchases, however, are meant to run out on Friday. The Pensions and Lifetime Savings Association, a trade body that represents the pension industry, urged the central bank on Tuesday to extend its purchases until the end of the month.

The near-daily expansion of the Bank of England’s rescue plan highlighted the challenges facing central banks in stamping out problems fueled by a once-in-a-generation increase in inflation and interest rates. It also raised questions about whether the BOE was providing the right medicine to address the problem.

The turmoil sparked fresh demands on Monday for pension funds to come up with cash to shore up LDIs, or liability-driven investments, derivative-based strategies that were meant to help match the money they owe to retirees over the long term.

LDIs were at the root of the bond selloff that prompted the BOE’s original intervention. Pension plans in late September saw a wave of margin calls after Prime Minister

Liz Truss’s

government announced large, debt-funded tax cuts that fueled an unprecedented bond-market selloff.

The BOE launched its original bond-purchase program on Sept. 28, but it only restored calm for a couple of days before selling resumed. An expansion of the program on Monday backfired, with yields again soaring higher.

The selloff on Monday was “very reminiscent of two weeks ago,” said

Simeon Willis,

chief investment officer of XPS, a company that advises pension plans.

LDI strategies use leveraged financial derivatives tied to interest rates to amplify returns. The outsize moves in U.K. bond markets last month led to huge collateral calls on pensions to back up the leveraged investments. The pension funds have sold other assets, including government and corporate bonds, to meet those calls, adding to pressure on yields to rise and creating a spiral effect on markets.

Pensions are typically big holders of inflation-linked government bonds, which help protect the plans from both inflation and interest-rate changes. But these weren’t eligible in the BOE’s bond-buying program until Tuesday.

The U.K. helped pioneer bonds with payouts linked to inflation, sometimes referred to as linkers, in the 1980s. Linkers were originally sold exclusively to pensions, but the U.K. opened them to other investors over the years.

Pensions remain a dominant force in the market because the bonds offer long-term protection against both inflation and interest-rate changes. Their outsize role left the market vulnerable to shifts in pension-fund demand like that seen in recent weeks.

Adam Skerry, a fund manager at Abrdn with a focus on inflation-linked government bonds, said his firm has struggled to trade those assets in recent days.

“We were trying to sell some bonds this morning, and it was virtually impossible to do that,” he said. “The LDI issue that’s facing the market, the fact that the market is moving to the degree that it did, particularly yesterday, suggests that there’s still an awful lot [of selling] there.”

Pensions have also appeared hesitant to sell their bonds to the BOE, reflecting a mismatch in what the central bank is offering and what the market needs.

“The way that the bank has structured this intervention is they can only buy assets if people put offers into them, but nobody is putting offers in,” said Craig Inches, head of rates and cash at Royal London Asset Management. He said the pension funds would rather sell their riskier assets, including corporate bonds or property.

Mr. Willis of XPS said many pensions want to hold on to their government bonds because it helps protect pensions against changes in interest rates, which impact the way their liabilities are valued.

“If they sell gilts now, they’re doing it in the likelihood that they’ll need to buy them back in the future at some point and they might be more expensive, and that’s unhelpful,” he said.

Also plaguing the program: Pension funds are traditionally slow-moving organizations that make decisions with multidecade horizons. The market turmoil has hurtled them into the warp-speed-style moves usually reserved for traders at swashbuckling hedge funds.

To make decisions about the sale of assets, industry players describe a game of telephone playing out among trustees, investment advisers, fund managers and banks. Pension funds spread their assets among multiple managers, which are in turn held by separate custodian banks. Calling everyone for the necessary signoffs is creating a lengthy and involved process.

To give themselves more time, pension funds are pushing the BOE to extend the bond-buying program at least to the end of the month. That is when the U.K.’s Treasury chief,

Kwasi Kwarteng,

is expected to lay out the government’s borrowing plans for the coming year.

The Institute for Fiscal Studies, a nonpartisan think tank that focuses on the budget, warned Tuesday that borrowing is likely to hit £200 billion in the financial year ending March, the third highest for a fiscal year since World War II and £100 billion higher than planned in March of this year. Increased borrowing increases the supply of bonds and generally causes bond yields to rise.

Mr. Kwarteng on Tuesday declared his confidence in BOE Gov. Andrew Bailey as he faced questions from lawmakers for the first time in his new job.

“I speak to the governor very frequently and he is someone who is absolutely independent and is managing what is a global situation very effectively,” he said.

—Caitlin Ostroff contributed to this article.

Write to Chelsey Dulaney at Chelsey.Dulaney@wsj.com, Anna Hirtenstein at anna.hirtenstein@wsj.com and Paul Hannon at paul.hannon@wsj.com